We consider entire solutions to Lu=f(u) in R2, where L is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator L, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.
A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane / F. Hamel, X. Ros Oton, Y. Sire, E. Valdinoci. - In: ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE. - ISSN 0294-1449. - 34:2(2017 Apr), pp. 469-482. [10.1016/j.anihpc.2016.01.001]
A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane
E. ValdinociUltimo
2017
Abstract
We consider entire solutions to Lu=f(u) in R2, where L is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator L, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.
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